We show: The procedure mentioned in the title is often impossible. It requires at least an inner model with a measurable cardinal. The consistency strength of changing b and d from a regular κ to some regular δ < κ is a measurable of Mitchell order δ. There is an application to Cichon's diagram.

We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem. Suppose κ is a singular strong limit cardinal and 2κ λ where λ is not the successor of a cardinal of cofinality at most κ. If cf > ω then it follows that o λ, and if cf = ωthen either o λ or {α: K o α+n} is confinal in κ for (...) each n ε ω.We also prove several results which extend or are related to this result, notably Theorem. If 2ω ω1 then there is a sharp for a model with a strong cardinal.In order to prove these theorems we give a detailed analysis of the sequences of indiscernibles which come from applying the covering lemma to nonoverlapping sequences of extenders. (shrink)

Reviewed Works:John R. Steel, A. S. Kechris, D. A. Martin, Y. N. Moschovakis, Scales on $\Sigma^1_1$ Sets.Yiannis N. Moschovakis, Scales on Coinductive Sets.Donald A. Martin, John R. Steel, The Extent of Scales in $L$.John R. Steel, Scales in $L$.

We prove for any $\mu = \mu^{ large enough (just strongly inaccessible Mahlo) the consistency of 2 μ = λ → [θ] 2 3 and even 2 μ = λ → [θ] 2 σ,2 for $\sigma . The new point is that possibly $\theta > \mu^+$.

LetSbe the group of all permutations of the set of natural numbers. The cofinality spectrumCF(S)ofSis the set of all regular cardinalsλsuch thatScan be expressed as the union of a chain ofλproper subgroups. This paper investigates which setsCof regular uncountable cardinals can be the cofinality spectrum ofS. The following theorem.is the main result of this paper.Theorem.Suppose that V ⊨ GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions.(a)C contains a maximum element.(b)Ifμis an inaccessible cardinal such (...) thatμ= sup(C∩μ),thenμ∈C.(c)ifμis a singular cardinal such thatμ= sup(C∩μ),thenμ+∈C.Then there exists a c.c.c. notion of forcing ℙ such that Vℙ⊨ CF(S) = C.We shall also investigate the connections between the cofinality spectrum andpcftheory; and show thatCF(S)cannot be an arbitrarily prescribed set of regular uncountable cardinals. (shrink)

We show: The procedure mentioned in the title is often impossible. It requires at least an inner model with a measurable cardinal. The consistency strength of changingandfrom a regularκto some regularδ<κis a measurable of Mitchell orderδ. There is an application to Cichoń's diagram.